Saddle point problem in Navier-Stokes equation

by Yi Zhang

provided that is symmetric and positive definite. Here which is a Hilbert space, and while is a bilinear operator.

This is a very general problem considered widely in many areas. However, in the real world, some further condition are to be applied on the space where lies on. If this is described as constraint equation

then we are dealing with constraint optimization problem ,to which Lagrange multiplier is usually the first one can come up. We then try to solve the stationary point of Lagrangian

Now supposing we set , using stationary point condition on above equation, we arrive at

where is replaced by indicating the parallel unknown as in the problem. The stationary point nature of this solution gives the name of saddle point problem, whose general case is

So, putting a constraint to a minimization problem, one try to solve a saddle problem instead of a simple linear system with SPD matrix.

For a linear system, one can always decompose it in to the form of

and this is the most general form of saddle problem.

For incompressible Navier-Stokes equation, we have (without mentioning the boundary conditions)